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Encyclopedia > Beth number

be the cardinality of countably infinite sets; for concreteness, take the set of natural numbers to be the typical case. Denote by P(A) the power set of A, i.e., the set of all subsets of A. Then define In mathematics the term countable set is used to describe the size of a set, e. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...

= the cardinality of the power set of A if is the cardinality of A.

Then

are respectively the cardinalities of

Each set in this sequence has cardinality strictly greater than the one preceding it, because of Cantor's theorem. Note that the 1st beth number is equal to c, the cardinality of the continuum, and the 2nd beth number is the power set of c. Note: in order to fully understand this article you may want to refer to the set theory portion of the table of mathematical symbols. ... In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In set theory and other branches of mathematics, ‭ב‬2 (pronounced beth two), or 2c (pronounced two to the power of c), is a certain cardinal number. ...

For infinite limit ordinals κ, we define A limit ordinal is an ordinal number which is not a successor ordinal. ...

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If we assume the axiom of choice, then infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable, and so, since no infinite cardinalities are between and , the celebrated continuum hypothesis can be stated in this notation by saying In mathematics, the axiom of choice, or AC, is an axiom of set theory. ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

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The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers. In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ... Results from FactBites:

 PlanetMath: beth numbers (110 words) The beth numbers are infinite cardinal numbers defined in a similar manner to the aleph numbers, as described below. Cross-references: uncountable, strong limit cardinal, generalized continuum hypothesis, inequality, cardinality of the continuum, limit ordinal, ordinal number, aleph numbers, cardinal numbers, infinite This is version 5 of beth numbers, born on 2004-04-02, modified 2006-09-13.
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